Im a college student in Mechanical Vibrations and this is by far the best and most thorough explanation i have ever seen of this topic. You explain the mechanics so eloquently.
@@madaydude_physics I might make my own simulation. I made a numerical damped pendulum simulation, but this would be really interesting. Do you think it'd make sense to make a "coil" shape by taking a sine function with a fixed number of periods and plotting it in space while shifting it and rotating it as the spring expands and contracts? Because I'm pretty sure a coil from a side view is just a trig function
@@hydropage2855 Yup, there are different periodic shapes people will use for their springs, but that’s the right idea. I’d be happy to see a video of your simulation if you end up making it :3
@@madaydude_physics What program did you use? I think I’ll use Processing. Also, I’m really curious, how can damping be incorporated into a Lagrangian? I’m not sure how damping would work for a spring-dulum in general, I’m struggling to imagine that
@@hydropage2855 Hi again hydro-- one nice way to incorporate damping is with the Rayleigh Dissipation Function: these links will explain the basics phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/10%3A_Nonconservative_Systems/10.04%3A_Rayleighs_Dissipation_Function#:~:text=The%20Rayleigh%20dissipation%20function%20R(q%2C%CB%99q)%20provides,both%20Lagrangian%20and%20Hamiltonian%20mechanics.&text=Consider%20the%20two%20identical%2C%20linearly,%CE%B2)%20shown%20in%20the%20figure. en.wikipedia.org/wiki/Rayleigh_dissipation_function
I’d love to see you take this a step further with a torsion-spring-pendulum sort of deal, if that makes sense? Take this spring-dulum here and apply some torque to it as well as set it in motion and stretches from spring equilibrium
Theorem 9: The Euler-Lagrange equations, which are the fundamental equations of motion in classical mechanics and field theory, can be derived from the principle of least action, which states that the path taken by a system between two points is the one that minimizes the action integral. Proof: Let q_i(t) be the generalized coordinates of a system, and let L(q_i, dq_i/dt, t) be the Lagrangian of the system, which is a function of the coordinates, their time derivatives, and time. The action integral S is defined as the integral of the Lagrangian over time: S = ∫_t1^t2 L(q_i, dq_i/dt, t) dt The principle of least action states that the path taken by the system between two points (q_i(t1), q_i(t2)) is the one that minimizes the action integral S. To find the equations of motion, we require that the variation of the action integral with respect to the path is zero: δS = 0 Using the calculus of variations, we can show that this condition leads to the Euler-Lagrange equations: (d/dt) (∂L/∂(dq_i/dt)) - (∂L/∂q_i) = 0 for each generalized coordinate q_i. These equations describe the motion of the system and can be used to derive the conservation laws and symmetry principles of classical mechanics and field theory. The fact that the equations of motion can be derived from a variational principle, which involves minimizing an integral, suggests that the concept of zero or nothingness (in the sense of a minimum or stationary point) may play a fundamental role in the dynamics of physical systems. Moreover, the action integral itself can be interpreted as a measure of the "amount of nothingness" in the path of the system, in the sense that it vanishes for the classical path (the one that satisfies the equations of motion) and is positive for all other paths. This interpretation suggests that the classical path of a system can be seen as a "zero mode" or "vacuum state" of the action integral, and that the properties of this zero mode may be related to the fundamental laws of physics and the symmetries of nature.
Hey! , a genuine question here from trigonometry. I was doing trig then i came upon an angle whose value in trig functions i forgot from the table, From their i remembered a trick of learning those values from the early days, to write down the numbers(for angles-(0,30,45,60,90 only) 0,1,2,3,4 then dividing these by 4 and then taking a square root and then respectively we get the values 0,1/2,1/rt2,rt3/2,1 *Why does this trick work?*i am getting an insight into this regarding the unit circle and the 4 quadrants but still cannot get an accurate answer , tried finding the answer on google, it was something like mentioned above but it did not explain well. Kindly spend a minute or two on this thought and if possible please make a video of it. Thankyou Amazing video by the way👍🏻👍🏻👍🏻👍🏻👍🏻👍🏻👍🏻
If the initial condition of the object is released from the angle θ and without initial velocity: a. What is the maximum spring elongation length? b. What is the speed of the object when θ = 0? Also what is the elongation of the spring at that time? How do you find the 2 points above?
a. Use conservation of energy law. I assume that the object is released from relaxed spring. Then, the total energy at the beginning is E0 = -mg l0 cos(θ). Maximim elongation happens when kinetic energy is zero and θ=0 (so all energy is used to elongate the spring) Then by energy conservation law, we get a quadratic equation: E0 = V + T = -mg(l0 + ρ) + 1/2 k ρ² + 0, or: -mg l0 cos(θ) = -mg(l0 + ρ) + 1/2 k ρ² which gives two solutions: 1. ρ = 0 2. ρ = 2mg/k Since we are looking for maximum elonagtion, the accept the second solution, ρ = 2mg/k. Note that this is an upper bound on elonagtion and might not be reached. But I suspect that unless there is some weird resonance, the spring will come arbitrary close to this elonagtion. b. I am almost sure that you cannot calculate this. You can only calculate pairs of speed and elonagtion that are possible. You could also do some asymptotic analysis to approximate the solution, if approximation is good enough for your application.
Hi, I think your videos on these topics are quite good. May I ask, what's your educational background/ can viewers expect videos anytime soon on topics like quantum field theory or relativity? Thanks again for your work
I’m glad you enjoy the videos here! I’m currently an undergraduate going into a Physics PhD, doing experiment based research, not theory, so generally speaking my videos are naturally going to have a bit more of a utilitarian flavor to them. To answer your question: it’s probable I will *eventually* cover such concepts, but regardless I would want to make more foundational content with undergrad level E&M, Quantum, Thermo etc before I get to that (assuming I don’t drown in grad level work and research first haha).
Is there (in your opinion) ever a case where it is advantageous to use polar coordinates q=(p,theta) as generalized coordinates as opposed to Cartesian coordinates q=(x,y). While the method you presented is generally found in textbooks, I found that derivations with Cartesian coordinates yield significantly faster simulations, and that Cartesian coordinates allows for considerably easier derivations for multi-spring systems, which in addition are trivial in Hamiltonian mechanics.
I don’t have a very strong opinion on this either way for this problem, as you would know from your simulation work formalisms like Lagrangian and Hamiltonian mechanics are so nice due to their form invariance under coordinate transformations (canonical transformations at least). I would at the very least say in problem solving if you save on the number of coordinates in a different coordinate system, you should use it. For example, for a simple pendulum it would be far more efficient to use a single angle instead of tracking both x and y (or having to write out the dependence of y on x using your constraint since you’d be using more coordinates than your degrees of freedom otherwise). But yeah, of course coordinate systems are a choice, we can convert between them easily as well etc etc. Cool simulations by the way, I checked a few out :)
@@madaydude_physics Alright, thanks for the input. I've been curious for a while why derivations (textbooks and otherwise) almost always use polar coordinates. The angle is indeed the simplest choice for the 1DOF simple pendulum, and also for the zero gravity case of the spring pendulum that reduces to 1DOF from conservation of angular momentum. For the simple pendulum, it is actually more efficient to use two Cartesian coordinates than a single angle (about a factor of 2 or 3 in C. Trigonometric functions are just that slow) - although this requires constraints that generally make derivations non-trivial. Of course you may disregard numerical performance for tasks involving analytical treatment. Thanks, you too. Apparently there was a reason why my feed picked up on spring videos just now.
@@Zymplectic Yes, thank you as well, having this nice numerics perspective with an idea of the differences between those computation times will be nice for others to note as well
Omg, I loved this the other day :) You did a beautiful job explaining (youve gained a fan). Personally, do you prefer Lagrangian or Hamiltonian Mechanics?
Thank you! Oooh, that’s really tough >.> Hamiltonian Mechanics really appears everywhere, particularly foundational in Quantum… so that’s hard to beat + I happen to like thinking in terms of phase space! Both are amazing though!
Hey madaydude, you helped me a lot with this video. i did some work on a simplified grasshopper landing model,if you can help with checking what i did that would be helpful 😊
Glad to hear it! Now I’m no expert on grasshopper landing *ahem* so I might not be of too much use, but I’d be curious to hear about your work, sounds interesting!
@@madaydude_physics oh, don’t worry too much, the model is simple. It’s almost acts as a three link manipulator, i am mainly concerned about my (derivation, kinematics, and how i add an input), thanks in advance, so where can i send you the file?
@@husamaltalhi8579 Ok, try emailing it to me: use the email attached on my channel page under channel details... I'm going to use you as my guinea pig also to make sure that's set up right ;)
![The Most Useful Curve in Mathematics [Logarithms]](/img/1.gif)
Im a college student in Mechanical Vibrations and this is by far the best and most thorough explanation i have ever seen of this topic. You explain the mechanics so eloquently.
Wow, what an honor, much appreciated! :)
Criminally underrated. Such a relaxing voice and style, and a great well-paced explainer. Instant subscribe
Thanks! Glad to hear you enjoy hydro :)
@@madaydude_physics I might make my own simulation. I made a numerical damped pendulum simulation, but this would be really interesting. Do you think it'd make sense to make a "coil" shape by taking a sine function with a fixed number of periods and plotting it in space while shifting it and rotating it as the spring expands and contracts? Because I'm pretty sure a coil from a side view is just a trig function
@@hydropage2855 Yup, there are different periodic shapes people will use for their springs, but that’s the right idea. I’d be happy to see a video of your simulation if you end up making it :3
@@madaydude_physics What program did you use? I think I’ll use Processing. Also, I’m really curious, how can damping be incorporated into a Lagrangian? I’m not sure how damping would work for a spring-dulum in general, I’m struggling to imagine that
@@hydropage2855 Hi again hydro-- one nice way to incorporate damping is with the Rayleigh Dissipation Function: these links will explain the basics
phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/10%3A_Nonconservative_Systems/10.04%3A_Rayleighs_Dissipation_Function#:~:text=The%20Rayleigh%20dissipation%20function%20R(q%2C%CB%99q)%20provides,both%20Lagrangian%20and%20Hamiltonian%20mechanics.&text=Consider%20the%20two%20identical%2C%20linearly,%CE%B2)%20shown%20in%20the%20figure.
en.wikipedia.org/wiki/Rayleigh_dissipation_function
I’d love to see you take this a step further with a torsion-spring-pendulum sort of deal, if that makes sense? Take this spring-dulum here and apply some torque to it as well as set it in motion and stretches from spring equilibrium
Bro your channel is a gem 💎.
Keep uploading such videos.
Thank you! Will do :)
Theorem 9: The Euler-Lagrange equations, which are the fundamental equations of motion in classical mechanics and field theory, can be derived from the principle of least action, which states that the path taken by a system between two points is the one that minimizes the action integral.
Proof:
Let q_i(t) be the generalized coordinates of a system, and let L(q_i, dq_i/dt, t) be the Lagrangian of the system, which is a function of the coordinates, their time derivatives, and time.
The action integral S is defined as the integral of the Lagrangian over time:
S = ∫_t1^t2 L(q_i, dq_i/dt, t) dt
The principle of least action states that the path taken by the system between two points (q_i(t1), q_i(t2)) is the one that minimizes the action integral S.
To find the equations of motion, we require that the variation of the action integral with respect to the path is zero:
δS = 0
Using the calculus of variations, we can show that this condition leads to the Euler-Lagrange equations:
(d/dt) (∂L/∂(dq_i/dt)) - (∂L/∂q_i) = 0
for each generalized coordinate q_i.
These equations describe the motion of the system and can be used to derive the conservation laws and symmetry principles of classical mechanics and field theory.
The fact that the equations of motion can be derived from a variational principle, which involves minimizing an integral, suggests that the concept of zero or nothingness (in the sense of a minimum or stationary point) may play a fundamental role in the dynamics of physical systems.
Moreover, the action integral itself can be interpreted as a measure of the "amount of nothingness" in the path of the system, in the sense that it vanishes for the classical path (the one that satisfies the equations of motion) and is positive for all other paths.
This interpretation suggests that the classical path of a system can be seen as a "zero mode" or "vacuum state" of the action integral, and that the properties of this zero mode may be related to the fundamental laws of physics and the symmetries of nature.
Hey! , a genuine question here from trigonometry.
I was doing trig then i came upon an angle whose value in trig functions i forgot from the table,
From their i remembered a trick of learning those values from the early days,
to write down the numbers(for angles-(0,30,45,60,90 only)
0,1,2,3,4 then dividing these by 4 and then taking a square root and then respectively we get the values
0,1/2,1/rt2,rt3/2,1
*Why does this trick work?*i am getting an insight into this regarding the unit circle and the 4 quadrants but still cannot get an accurate answer , tried finding the answer on google, it was something like mentioned above but it did not explain
well.
Kindly spend a minute or two on this thought and if possible please make a video of it.
Thankyou
Amazing video by the way👍🏻👍🏻👍🏻👍🏻👍🏻👍🏻👍🏻
Nice question, I think this is a great idea to make a little video on those proofs. I’ll add this to my video plans
I am just a grade 11th student but I enjoyed watching your video a lot! You gained another sub
Thanks, glad to hear it :)
If the initial condition of the object is released from the angle θ and without initial velocity:
a. What is the maximum spring elongation length?
b. What is the speed of the object when θ = 0? Also what is the elongation of the spring at that time?
How do you find the 2 points above?
a. Use conservation of energy law. I assume that the object is released from relaxed spring. Then, the total energy at the beginning is E0 = -mg l0 cos(θ). Maximim elongation happens when kinetic energy is zero and θ=0 (so all energy is used to elongate the spring)
Then by energy conservation law, we get a quadratic equation:
E0 = V + T = -mg(l0 + ρ) + 1/2 k ρ² + 0,
or:
-mg l0 cos(θ) = -mg(l0 + ρ) + 1/2 k ρ²
which gives two solutions:
1. ρ = 0
2. ρ = 2mg/k
Since we are looking for maximum elonagtion, the accept the second solution, ρ = 2mg/k.
Note that this is an upper bound on elonagtion and might not be reached.
But I suspect that unless there is some weird resonance, the spring will come arbitrary close to this elonagtion.
b. I am almost sure that you cannot calculate this. You can only calculate pairs of speed and elonagtion that are possible. You could also do some asymptotic analysis to approximate the solution, if approximation is good enough for your application.
Hi, I think your videos on these topics are quite good. May I ask, what's your educational background/ can viewers expect videos anytime soon on topics like quantum field theory or relativity? Thanks again for your work
I’m glad you enjoy the videos here! I’m currently an undergraduate going into a Physics PhD, doing experiment based research, not theory, so generally speaking my videos are naturally going to have a bit more of a utilitarian flavor to them. To answer your question: it’s probable I will *eventually* cover such concepts, but regardless I would want to make more foundational content with undergrad level E&M, Quantum, Thermo etc before I get to that (assuming I don’t drown in grad level work and research first haha).
Is there (in your opinion) ever a case where it is advantageous to use polar coordinates q=(p,theta) as generalized coordinates as opposed to Cartesian coordinates q=(x,y).
While the method you presented is generally found in textbooks, I found that derivations with Cartesian coordinates yield significantly faster simulations, and that Cartesian coordinates allows for considerably easier derivations for multi-spring systems, which in addition are trivial in Hamiltonian mechanics.
I don’t have a very strong opinion on this either way for this problem, as you would know from your simulation work formalisms like Lagrangian and Hamiltonian mechanics are so nice due to their form invariance under coordinate transformations (canonical transformations at least). I would at the very least say in problem solving if you save on the number of coordinates in a different coordinate system, you should use it. For example, for a simple pendulum it would be far more efficient to use a single angle instead of tracking both x and y (or having to write out the dependence of y on x using your constraint since you’d be using more coordinates than your degrees of freedom otherwise). But yeah, of course coordinate systems are a choice, we can convert between them easily as well etc etc. Cool simulations by the way, I checked a few out :)
@@madaydude_physics Alright, thanks for the input. I've been curious for a while why derivations (textbooks and otherwise) almost always use polar coordinates.
The angle is indeed the simplest choice for the 1DOF simple pendulum, and also for the zero gravity case of the spring pendulum that reduces to 1DOF from conservation of angular momentum.
For the simple pendulum, it is actually more efficient to use two Cartesian coordinates than a single angle (about a factor of 2 or 3 in C. Trigonometric functions are just that slow) - although this requires constraints that generally make derivations non-trivial. Of course you may disregard numerical performance for tasks involving analytical treatment.
Thanks, you too. Apparently there was a reason why my feed picked up on spring videos just now.
@@Zymplectic Yes, thank you as well, having this nice numerics perspective with an idea of the differences between those computation times will be nice for others to note as well
Omg, I loved this the other day :) You did a beautiful job explaining (youve gained a fan). Personally, do you prefer Lagrangian or Hamiltonian Mechanics?
Thank you! Oooh, that’s really tough >.> Hamiltonian Mechanics really appears everywhere, particularly foundational in Quantum… so that’s hard to beat + I happen to like thinking in terms of phase space! Both are amazing though!
Waw is good
Hey madaydude, you helped me a lot with this video.
i did some work on a simplified grasshopper landing model,if you can help with checking what i did that would be helpful 😊
Glad to hear it! Now I’m no expert on grasshopper landing *ahem* so I might not be of too much use, but I’d be curious to hear about your work, sounds interesting!
@@madaydude_physics oh, don’t worry too much, the model is simple. It’s almost acts as a three link manipulator, i am mainly concerned about my (derivation, kinematics, and how i add an input), thanks in advance, so where can i send you the file?
@@husamaltalhi8579 Ok, try emailing it to me: use the email attached on my channel page under channel details... I'm going to use you as my guinea pig also to make sure that's set up right ;)
@@madaydude_physics i sent the files, 🙏
@@husamaltalhi8579 Excellent, I will check it out when I have spare time